**3. Lie Group Framework**

∂*y* Obtaining the solutions of the PDEs (partial differential equations) (10)–(12) governing the investigation understudy is equivalent to satisfying the constant solutions of these equations under a special continuous one-parameter group. The proposed technique is to search for a transformation group from the primary collection of one parameter scaling transformation. The facilitated form of Lie group framework, namely, the scaling group of transformations Δ (see [38–46]), will be presented here:

$$
\Lambda \colon \widehat{\bf x}^\* = x \ell^{\mathrm{ex}\_1}, \widehat{\bf y}^\* = y \ell^{\mathrm{ex}\_2}, \widehat{\bf t} = t \ell^{\mathrm{ex}\_3}, \widehat{\bf Y} = \Psi \ell^{\mathrm{ex}\_4}, \overline{\theta} = \theta \ell^{\mathrm{ex}\_5} \tag{13}
$$

where κ1, κ2, κ3, κ4, and κ5 are transformation parameters and ε is a small parameter whose interrelationship will be determined by our investigation. Equation (13) may be scrutinized as a point transformation, which transfers the coordinates (*x*, *y*, *t*, Ψ, θ) to ( *x* , *y*, *t* , Ψ, <sup>θ</sup>). Substituting transformations Equation (13) in Equations (10)–(12), we obtain;

$$\begin{split} \ell^{\varepsilon(\mathbf{x}\_{2}+\mathbf{x}\_{3}-\mathbf{x}\_{4})} & \frac{\partial^{2}\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{i}}\ \widehat{\mathbf{j}}} + \ell^{\varepsilon(\mathbf{x}\_{1}+2\mathbf{x}\_{2}-2\mathbf{x}\_{4})} \left( \frac{\partial\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{y}}} \frac{\partial^{2}\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{x}}\ \widehat{\mathbf{j}}} - \frac{\partial\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{x}}} \frac{\partial^{2}\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{y}}^{2}} \right) &= \nu\_{f}\Xi\_{1}\ell^{\varepsilon(3\mathbf{x}\_{2}-\mathbf{x}\_{4})} \frac{\partial^{3}\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{y}}^{3}} \\ -\frac{\sigma\_{ff}}{\sigma\_{f}} \frac{\partial^{2}\sigma\_{f}}{\partial\rho} \frac{1}{1-\phi+\phi(\rho\_{s}/\rho\_{f})} \ell^{\varepsilon(\mathbf{x}\_{2}-\mathbf{x}\_{4})} \frac{\partial\widehat{\mathbf{v}}}{\partial\widehat{\mathbf{y}}} \end{split} \tag{14}$$

$$\mathcal{E}^{\varepsilon(\mathbf{x}\_3-\mathbf{x}\_5)} \frac{\partial \widehat{\partial}}{\partial \widehat{\ }} + \mathcal{E}^{\varepsilon(\mathbf{x}\_1+\mathbf{x}\_2-\mathbf{x}\_4-\mathbf{x}\_5)} \left( \frac{\partial \widehat{\mathbf{v}}}{\partial \widehat{\ }} \frac{\partial \widehat{\partial}}{\partial \widehat{\ }} - \frac{\partial \widehat{\mathbf{v}}}{\partial \widehat{\ }} \frac{\partial \widehat{\partial}}{\partial \widehat{\ }} \right) = \frac{\upsilon\_f}{\text{Pr}} \mathbb{E} \mathbf{z} \Big( \frac{k\_{ff}}{k\_f} + \frac{4}{3} \text{Rd} \Big) \mathcal{E}^{\varepsilon(2\mathbf{x}\_2-\mathbf{x}\_5)} \frac{\partial^2 \widehat{\partial}}{\partial \widehat{\ }} \tag{15}$$

The following relations should be determined to reserve the system to be constant:

$$\begin{aligned} \mathbf{x}\_2 + \mathbf{x}\_3 - \mathbf{x}\_4 &= \mathbf{x}\_1 + 2\mathbf{x}\_3 - 2\mathbf{x}\_4 = 3\mathbf{x}\_2 - \mathbf{x}\_4 = \mathbf{x}\_2 - \mathbf{x}\_4\\ \mathbf{x}\_3 + \mathbf{x}\_5 &= \mathbf{x}\_1 + \mathbf{x}\_2 - \mathbf{x}\_4 - \mathbf{x}\_5 = 2\mathbf{x}\_2 - \mathbf{x}\_5 \end{aligned} \tag{16}$$

These relations give

$$
\kappa\_4 = \kappa\_1, \kappa\_2 = \kappa\_3 = \kappa\_5 = 0 \tag{17}
$$

and the one-parameter group of transformations can be obtained as

$$
\widehat{\mathbf{x}} = \mathbf{x}\ell^{\mathbf{x}\mathbf{x}\_{1}}, \widehat{\mathbf{y}} = \mathbf{y}, \widehat{\mathbf{t}} = \mathbf{t}, \widehat{\mathbf{y}} = \mathbf{Y}\ell^{\mathbf{x}\mathbf{x}\_{1}}, \widehat{\boldsymbol{\theta}} = \boldsymbol{\theta} \tag{18}
$$

Developing by Taylor's technique in powers of ε, we obtain:

$$
\widehat{\mathbf{x}} - \mathbf{x} = \mathbf{x} \varepsilon \mathbf{x}\_1,\\
\widehat{\mathbf{y}} - \mathbf{y} = \mathbf{0},\\
\widehat{t} - t = \mathbf{0},\\
\widehat{\mathbf{y}} - \mathbf{y} = \mathbf{y} \varepsilon \mathbf{x}\_1,\\
\widehat{\boldsymbol{\theta}} - \boldsymbol{\theta} = \mathbf{0}
$$

which yields

$$\frac{d\mathbf{x}}{d\mathbf{x}\_1} = \frac{d\mathbf{y}}{0} = \frac{dt}{0} = \frac{d\Psi}{\Psi \mathbf{x}\_1} = \frac{d\theta}{0} \tag{19}$$

$$
\eta = \Gamma\_1(\mathbf{x}, t) \\
y, \tau = \Gamma\_2(\mathbf{x}, y) \\
t\_\prime \Psi = \Gamma\_3(y, t) \\
\mathbf{x}, \theta = \theta(\tau, \eta), \tag{20}
$$

where Γ1, Γ2, and Γ3 are arbitrary functions which should be determined by its equations. η and τ are the similarity variable and dimensionless time.

To avert the fluid properties manifesting explicitly in the coe fficients of the above equations, determining mass balance in Equation (1), with keeping generality, we have dropped three di fferent convenient arbitrary constants based on the transformations performed previously by Nabwey [25] and Chamkha [29] as follows:

$$
\Gamma\_1(\mathbf{x}, t) \,\,= \left(1/2a\sqrt{\nu\_f t}\right)\Gamma\_2(y, t) \,\,=\, a,\\
\Gamma\_3(y, t) \,\,=\, 2a\sqrt{\nu\_f t} \,\,\,\tag{21}
$$

As a consequence, we find

$$t = \frac{\pi}{a'} y = 2\sqrt{\nu\_f t \eta\_\prime} \Psi = 2ax\sqrt{\nu\_f t} f(\pi, \eta) \tag{22}$$

with the assistance of these formulations in Equation (22). Equations (10)–(12) are characterized as

$$2\nabla\_1 f'''' + 2\eta f'' - 4\pi \left( f'^2 - f f'' + \frac{\sigma\_{ff}}{\sigma\_f} \frac{H a^2}{1 - \phi + \phi(\rho\_s/\rho\_f)} f' \right) - 4\pi \frac{\partial f'}{\partial \tau} = 0 \tag{23}$$

$$2\frac{\Xi\_2}{\Pr} \left(\frac{k\_{ff}}{k\_f} + \frac{4}{3}R\right) \theta'' + 2\eta \theta' + 4\tau (f\theta' - f'\theta) - 4\tau \frac{\partial \theta}{\partial \tau} = 0\tag{24}$$

subject to the following boundary conditions:

$$\begin{cases} f(\tau,0) = 0, f'(\tau,0) = 1 + \frac{\delta/\sqrt{\tau}}{(1-\chi)^{2.5}} f''(\tau,0), \frac{k\_{ff}^{\prime}}{k\_f} \theta'(\tau,0) = -Bi\sqrt{\tau}(1-\theta(\tau,0))\\ f'(\tau,\infty) = 0, \theta(\tau,\infty) = 0 \end{cases} \tag{25}$$

where *Ha* = *B*0 + <sup>σ</sup>*f <sup>a</sup>*ρ*f* stands for the Hartmann number. *Bi* = 2*hf k f* +ν*f a* stands for Biot number. *Rd* = <sup>4</sup>σ1*T*<sup>3</sup> ∞/*k f* β*R* stands for the radiation parameter. δ = *<sup>L</sup>*μ*f* /2 + <sup>υ</sup>*f* /*a* stands for the velocity slip parameter.

The local skin-drag coe fficient and local Nusselt number can be written respectively, as

$$\mathcal{C}\_{f} = \left. -\mu\_{ff} \frac{\partial u}{\partial y} \right|\_{y=0} / \left( \mu\_{f} a \text{x} / 2 \sqrt{\nu\_{f} t} \right) = \left. -\frac{1}{\left( 1 - \chi \right)^{2.5}} f'' \left( \tau, 0 \right) \right|\_{y=0} \tag{26}$$

$$\begin{split} \mathcal{N}u &= -\Big[ \Big( k\_{ff} + \frac{16v\_{l}T\_{\infty}^{3}}{3\overline{\theta}\_{R}} \Big) \frac{\partial T}{\partial y} \Big]\_{y=0} / \Big( k\_{f}(T\_{f} - T\_{\infty}) / 2\sqrt{v\_{f}t} \Big) \\ &= -\Big( \frac{k\_{ff}}{k\_{f}} + \frac{4Rd}{3} \Big) \theta'(\tau, 0) \end{split} \tag{27}$$
